In Chow et al: The Ricci Flow [...] Part II: [...], MSM 144, appendix F
www.math.ucsd.edu/~benchow/TensorCalculusFrameBundle.pdf
it is stated on p. 412:
Our convention shall bee that GL(n,R) acts on the left on FM.
where FM is the (first order) linear frame bundle on a real smooth manifold M of dimension n. Then it is claimed on p. 413:
If $Y=(Y_a)_{a=1}^n$ and $Z=(Z_b)_{b=1}^n$ are frames [at $x \in M$] with $Z_b = G^a_b\, Y_a$ where $G = (G^a_b)^n_{a,b=1} \in GL(n,R)$, and [...] if W is a vector field on M, then $W^b(Z) = (G^{-1})_a^b\, W^a(Y)$.
On p.414, it is written, that if $x^i$ are local coordinates on M and $\frac{\partial}{\partial x^i}$ is the corresponding coordinate frame, then define the functions $y^i_a$ via
$Y_a = y^i_a(Y) \frac{\partial}{\partial x^i}$
and
The function $y^i_a$ assigns to a frame Y the i-the component of the a-th vector $Y_a$. [This is correct.] The vector- (or matrix-) valued function $y := (y^i_a)_{i,a=1}^n$ [...] describes the transition of the frame $(\frac{\partial}{\partial x^i})_{i=1}^n$ to the frame $ Y = (Y_a)_{a=1}^n$. That is, $(y^i_a)$ are the components of the frame $\frac{\partial}{\partial x^i}$ with respect to the frame $(Y_a)$.
I think, this appendix F is inconsistent. The formulas for the components are written as if one had the usual right action of GL(n,R) on the linear frame bundle FM, where a frame $(Y_a)$ at x is identified with an element Y of $Iso_{\mathbb{R}-Vec\,}\,(\mathbb{R}^n, T_xM)$, an R-linear isomorphism of vector spaces, which maps the a-th vector of the standard basis in R^n to the a-th frame vector $Y_a$, and the right group action is simply composition of Y and $G \in GL(n,R)$,
$R^n \stackrel{G}{\longrightarrow} R^n \stackrel{Y}\longrightarrow T_xM$.
So, the three (partially) cited statements should be written:
Our convention shall bee that GL(n,R) acts on the right on FM.
If $Y=(Y_a)_{a=1}^n$ and $Z=(Z_b)_{b=1}^n$ are frames [at $x \in M$] with $\boldsymbol{ Z_b = Y_a\, G^a_b}$ where $G = (G^a_b)^n_{a,b=1} \in GL(n,R)$, and [...]
The function $y^i_a$ assigns to a frame Y the i-the component of the a-th vector $Y_a$. The vector- (or matrix-) valued function $y := (y^i_a)_{i,a=1}^n$ [...] has as its components the components of the a-th frame vector $Y_a$ with respect to the frame $\frac{\partial}{\partial x^i}$.
Similarly, several formulas and text from section 4 on in appendix F should be corrected.
Am I right with this observation (and the authors were wrong, their formulas would not be consistent with a left group action), or do I miss some important point, maybe a very special convention? Thank you very much.